CSIAM Trans. Appl. Math., 6 (2025), pp. 593-624.
Published online: 2025-09
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The time-space nonlocal evolution equations are powerful implementation
for modeling anomalous diffusion. In this research, we study the nonlocal nonautonomous reaction-diffusion equation
where $\chi$ is a Lusin space, $∂^w_t$ is a generalized time fractional derivative, $κ$ is a bounded
reaction rate, and $\mathcal{L}$ is an infinitesimal generator in terms of semigroup induced by
a symmetric Markov process $X.$ We show that the stochastic representation $u(t,x)$ defined by
is the unique mild as well as weak solution. By further analysis, one can get that the
above stochastic representation is also the unique strong solution, and the higher spatial and temporal regularity are obtained. In some particular cases, the corresponding
dynamical behaviors are displayed.
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The time-space nonlocal evolution equations are powerful implementation
for modeling anomalous diffusion. In this research, we study the nonlocal nonautonomous reaction-diffusion equation
where $\chi$ is a Lusin space, $∂^w_t$ is a generalized time fractional derivative, $κ$ is a bounded
reaction rate, and $\mathcal{L}$ is an infinitesimal generator in terms of semigroup induced by
a symmetric Markov process $X.$ We show that the stochastic representation $u(t,x)$ defined by
is the unique mild as well as weak solution. By further analysis, one can get that the
above stochastic representation is also the unique strong solution, and the higher spatial and temporal regularity are obtained. In some particular cases, the corresponding
dynamical behaviors are displayed.
The time-space nonlocal evolution equations are powerful implementation
for modeling anomalous diffusion. In this research, we study the nonlocal nonautonomous reaction-diffusion equation
where $\chi$ is a Lusin space, $∂^w_t$ is a generalized time fractional derivative, $κ$ is a bounded
reaction rate, and $\mathcal{L}$ is an infinitesimal generator in terms of semigroup induced by
a symmetric Markov process $X.$ We show that the stochastic representation $u(t,x)$ defined by
is the unique mild as well as weak solution. By further analysis, one can get that the
above stochastic representation is also the unique strong solution, and the higher spatial and temporal regularity are obtained. In some particular cases, the corresponding
dynamical behaviors are displayed.